American Mathematical Monthly - February 1999

FEBRUARY 1999

Does Mathematics Need New Axioms?
by Solomon Fefermansf@csli.stanford.edu
From the time of his stunning incompleteness results in 1931 until the end of his life, Kurt Gödel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article "What is Cantor's continuum problem?" (as it happens, in the American Mathematical Monthly) he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Gödel's program, but there are considerable differences of opinion as to what conclusions to draw from their results. The history of that work is traced from the beginning axiomatizations of number theory by Dedekind and Peano, and of set theory by Zermelo and Fraenkel, to the very present. We then turn to an examination of what axioms of higher set theory are needed to settle problems in finite combinatorics, the continuum problem, and scientifically applicable mathematics, and close with some controversial conclusions.

Statistical Independence and Normal Numbers: An Aftermath to Mark Kac's Carus Monograph
by Gerald S. Goodmangoodman@ds.unifi.it
In his Carus Monograph on Statistical Independence, Kac gave a remarkably simple proof of the Strong Law of Large Numbers for Bernoulli trials by computing the fourth moments of sums of the first n Rademacher functions and appealing to Beppo Levi's Theorem. We replace Rademacher functions by their finite products, known as Walsh functions, and show that the same method leads to a corresponding limit theorem for the relative frequencies of finite strings of binary digits and, thus, to Borel's Normal Number Theorem in base 2.

Mend&eacutes-France generalized the Rademacher and Walsh functions to arbitrary integral bases b>1 by allowing them to take values in the cyclotomic group of order b. He then employed them to give a new proof of the Normal Number Theorem in base b. We obtain the same result in a more elementary way by applying Kac's method to these generalized functions.

Behind these results is the "multiplicativity" property of the Rademacher functions, which, in the general case, takes the form of the vanishing of their mixed moments. Pursuing an idea of R&eacutenyi's, we show that multiplicativity is equivalent to statistical independence of the b-adic digits. Since multiplicativity can be established directly, by calculus, we are led to a new proof of statistical independence. The same technique can also be used to establish statistical independence of certain subclasses of generalized Walsh functions.

The Velocity Dependence of Aerodynamic Drag: A Primer for Mathematicians
by Lyle N. Long and Howard Weisslnl@psu.edu, weiss@math.psu.edu
Many elementary ordinary differential equation and calculus texts contain a model of the motion of a body (like a baseball or skydiver) subject to a drag force that depends linearly on the velocity, along with a calculation of the body's terminal velocity. Unfortunately, the physical assumption regarding the linear dependence of the drag force on velocity is often incorrect, and thus the model's predictions are physically implausible. The purpose of this note is to explain the dependence of the drag force on velocity for a general mathematical audience and to present a few realistic models. The appendix contains an interesting model (which has a closed-form solution) of re-entry of the space shuttle into the earth's atmosphere.

George Green: An Enigmatic Mathematician
by D. M. Cannellppzlaw@ppn1.nottingham.ac.uk
George Green was an early nineteenth-century mathematician and physicist, whose work is widely known but whose personality has remained a mystery. This paper attempts to dispel what has been termed "the miasmal mist surrounding Green's life." It examines the social and personal circumstances of a working miller, struggling to publish his first and seminal work, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (in which he introduced the eponymous functions and theorem), and which Einstein declared was twenty years ahead of its time. Some puzzling aspects of Green's life are examined, e.g., his lack of formal education, his access to continental sources, his posthumous neglect. It discusses the delicate relationship between Green and his aristocratic patron, and traces the events that led to Green's ultimate recognition as one of the most important English mathematicians. For more information about George Green, mathematician and physicist, visit http://www.nottingham.ac.uk/~ppzwww/green/.

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